How do you find the Vertical, Horizontal, and Oblique Asymptote given #x /( 4x^2+7x-2)#?

Question

Benjamin Achtenberg · Accepted Answer

vertical asymptotes at x = -2 , x #=1/4#
horizontal asymptote at y = 0

The denominator of the function cannot be zero as this would make the function undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes. solve: #4x^2+7x-2=0rArr(4x-1)(x+2)=0# #rArrx=-2" and " x=1/4" are the asymptotes"# Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" (a constant)"# divide terms on numerator/denominator by the highest power of x that is #x^2# #f(x)=(x/x^2)/((4x^2)/x^2+(7x)/x^2-2/x^2)=(1/x)/(4+7/x-2/x^2)# as #xto+-oo,f(x)to0/(4+0-0)# #rArry=0" is the asymptote"# Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 1 ,denominator-degree 2 ) Hence there are no oblique asymptotes. graph{x/(4x^2+7x-2) [-10, 10, -5, 5]}

Christian Antunez · Answer

undefined To find the vertical asymptotes, set the denominator equal to zero and solve for x. The values of x obtained are the vertical asymptotes.

To find the horizontal asymptote, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of both. The resulting value is the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

To find the oblique asymptote, perform polynomial long division or synthetic division. The quotient obtained is the equation of the oblique asymptote. If the degree of the numerator is less than the degree of the denominator, the oblique asymptote is y = 0. If the degree of the numerator is greater than the degree of the denominator by exactly 1, there is an oblique asymptote.

For the given function x/(4x^2 + 7x - 2):

1. Vertical asymptotes: Set the denominator equal to zero and solve for x. The values obtained are the vertical asymptotes.

2. Horizontal asymptote: Compare the degrees of the numerator and denominator. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

3. Oblique asymptote: Perform polynomial long division or synthetic division. The quotient obtained is the equation of the oblique asymptote. If the degree of the numerator is less than the degree of the denominator, the oblique asymptote is y = 0. If the degree of the numerator is greater than the degree of the denominator by exactly 1, there is an oblique asymptote.